# 1.5: Dividing Fractions

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

When dividing fractions you invert (flip upside down) the fraction on the right side of the equation (the dividend). Then it becomes a multiplication problem. Invert and multiply!

Example $$\PageIndex{1}$$

$$\dfrac{2}{5} \div \dfrac{1}{2} \rightarrow \dfrac{2}{5} \times \dfrac{2}{1}=\dfrac{4}{5}$$

Example $$\PageIndex{2}$$

$$\dfrac {\dfrac{4}{9}}{\dfrac{8}{9}}$$

This example reads $$\dfrac{4}{9}$$ divided by $$\dfrac{8}{9}$$.

However, after you “invert and multiply,” it becomes:

$\dfrac{4}{9} \times \dfrac{9}{8} \rightarrow \dfrac{1}{1} \times \dfrac{1}{2}=\dfrac{1}{2} \nonumber$

After inverting the fraction, the same rules apply as previously mentioned when multiplying fractions. You need to change mixed numbers into improper fractions; you can cross cancel, and always remember to reduce if necessary.

Example $$\PageIndex{3}$$

$$\dfrac{3}{8} \div \dfrac{12}{6} \rightarrow \dfrac{3}{8} \times \dfrac{6}{12} \rightarrow \dfrac{\not{3}}{\not {8}} \times \dfrac{\not {6}}{\not {12}}=\dfrac{1}{4} \times \dfrac{3}{4}=\dfrac{3}{16}$$

In the Example $$\PageIndex{3}$$ above the 3 divided into itself once and into the 12 four times. Similarly, 2 divided into 6, three times and into 8, four times. Can you see a different way to cross cancel?

If the dividend is a whole number write it as a fraction before inverting. Always remember to cross cancel and reduce if necessary.

Example $$\PageIndex{4}$$

$$\dfrac{5}{8} \div 10 \rightarrow \dfrac{5}{8} \div \dfrac{10}{1} \rightarrow \dfrac{5}{8} \times \dfrac{1}{10} \rightarrow \dfrac{\not{5}}{8} \times \dfrac{1}{\not {10}} \rightarrow \dfrac{1}{8} \times \dfrac{1}{2}=\dfrac{1}{16}$$

## Exercise 1.5

Divide the following and reduce if necessary.

1. $$\dfrac{1}{2} \div \dfrac{2}{4}$$
2. $$\dfrac{5}{22} \div 7 \dfrac{2}{4}$$
3. $$\dfrac{6}{8} \div \dfrac{9}{12}$$
4. $$3 \dfrac{1}{3} \div 10$$
5. $$5 \dfrac{1}{4} \div 8 \dfrac{1}{2}$$
6. $$\dfrac{8}{16} \div \dfrac{16}{8}$$
7. $$4 \div 3 \dfrac{2}{3}$$
8. $$\dfrac{2}{5} \div 5 \dfrac{6}{9}$$
9. $$\dfrac{9}{13} \div 9$$
10. $$\dfrac{11}{22} \div \dfrac{1}{2}$$
11. $$2 \dfrac{9}{20} \div 5 \dfrac{2}{5}$$
12. $$3 \dfrac{1}{2} \div 9 \dfrac{1}{2}$$
13. $$5 \div 25$$
14. $$\dfrac{10}{3} \div \dfrac{1}{6}$$
15. $$\dfrac{13}{33} \div \dfrac{39}{3}$$
16. $$100 \dfrac{1}{2} \div 10 \dfrac{5}{6}$$

1.5: Dividing Fractions is shared under a CC BY license and was authored, remixed, and/or curated by LibreTexts.